Two-photon-exchange effects in the unpolarized $\mu p$ scattering within a hadronic model

In this work, the two-photon-exchange (TPE) effects in the unpolarized $\mu p$ scattering are discussed within the hadronic model where the intermediate states $N,\mathrm{\Delta }$, and $\sigma$ are considered. The contribution from the $N$ intermediate is close to the results given by Tomalak and Vanderhaeghen [Phys. Rev. D 90, 013006 (2014)] at small $Q$, and there is a sizable difference when $Q>0.25\mathrm{GeV}$ (where $Q^2$ is the four-momentum transfer). The contributions from the $\mathrm{\Delta }$ and the $\sigma$ intermediate states are much smaller than that from the $N$ intermediate at small $Q$. In the kinematic region with $k_i\subseteq [0.01,0.3]$ GeV and $Q\le 0.4\mathrm{GeV}$ (where $k_i$ is the three-momentum of initial muon in the laboratory frame), a naive expression for the TPE contributions is given, which can be used directly for other analysis.

Figure 1

One-photon-exchange diagram for $\mu p$ scattering in the Feynman gauge.

Figure 2

TPE contributions to $\mu p$ scattering from the $s,u$ channels owing to the $N,\mathrm{\Delta }$ intermediate states in Feynman gauge.

Figure 3

$\sigma$ meson exchange between muons and protons with photon and pion loops, (e) boxlike diagram; (f) crossed-box-like diagram; (g) contactlike diagram.

Figure 4

$\sigma$ meson exchange between muons and protons with photon and proton loops, (h) boxlike diagram; (i) crossed-box-like diagram.

Figure 5

TPE corrections from the $N$ intermediate state ${\delta }_{\mu p}^{\left(N\right)}$ as a function of $Q$ at $k_i=0.115,0.158,0.21$ GeV, with $k_i$ the three-momentum of the initial muon in the laboratory frame. (a) The corrections using Eq. (13) as input; (b) the corrections using Eq. (18) of Ref. [10] as input.

Figure 6

TPE corrections from the $\mathrm{\Delta }$ intermediate state ${\delta }_{\mu p}^{\left(\mathrm{\Delta }\right)}\text{vs}Q$ at $k_i=0.115,0.158,0.21\mathrm{GeV}$ with $k_i$ the three-momentum of the initial muon in the laboratory frame.

Figure 7

TPE corrections from the $\sigma$ intermediate state ${\delta }_{\mu p}^{(\sigma ,\pi +N)}$ at $k_i=0.115,0.158,0.21\mathrm{GeV}$ with $k_i$ the three-momentum of the initial muon in the laboratory frame. (a) ${\delta }_{\mu p}^{(\sigma ,\pi +N)}\text{vs}Q$; (b) ${\delta }_{\mu p}^{(\sigma ,\pi +N)}\text{vs}{\theta }_{\text{lab}}$.

Figure 8

Numerical results for $g_{\sigma \mu \mu }\text{vs}{Q}^2$, which can be compared directly with the $f_s$ in Ref. [31]; ppm means ${10}^{-6}$.